MinT - Best Known (251−132, 251, s)-Nets in Base 2

Best Known (251−132, 251, s)-Nets in Base 2

(251−132, 251, 57)-Net over F₂ — Constructive and digital

Digital (119, 251, 57)-net over F₂, using

t-expansion [i] based on digital (110, 251, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(251−132, 251, 73)-Net over F₂ — Digital

Digital (119, 251, 73)-net over F₂, using

t-expansion [i] based on digital (114, 251, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(251−132, 251, 249)-Net over F₂ — Upper bound on s (digital)

There is no digital (119, 251, 250)-net over F₂, because

12 times m-reduction [i] would yield digital (119, 239, 250)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²³⁹, 250, F₂, 120) (dual of [250, 11, 121]-code), but
  - residual code [i] would yield linear OA(2¹¹⁹, 129, F₂, 60) (dual of [129, 10, 61]-code), but
    - residual code [i] would yield linear OA(2⁵⁹, 68, F₂, 30) (dual of [68, 9, 31]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁶⁰, 69, F₂, 31) (dual of [69, 9, 32]-code), but
        â€œBGVâ€ bound on codes from Brouwerâ€™s database [i]

(251−132, 251, 251)-Net in Base 2 — Upper bound on s

There is no (119, 251, 252)-net in base 2, because

6 times m-reduction [i] would yield (119, 245, 252)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴⁵, 252, S₂, 126), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 7237 005577 332262 213973 186563 042994 240829 374041 602535 252466 099000 494570 602496 / 127 > 2²⁴⁵ [i]